Self-similarity symmetry and fractal distributions in iterative dynamics of dissipative mappings

TL;DR

This paper explores how iterative dynamical mappings of signals can produce fractal distributions and self-similarity, with applications to chaotic systems and dissipative circuits, revealing steady states governed by dilatation equations.

Contribution

It introduces the concept that steady states in iterative mappings can be characterized by fractal distributions and dilatation equations, extending understanding of fractal behavior in dynamical systems.

Findings

Steady states satisfy dilatation equations relating functions at different scales

Both linear and nonlinear chaotic systems exhibit fractal distributions

Dissipative circuits with delayed feedback show fractal self-similarity

Abstract

We consider transformations of deterministic and random signals governed by simple dynamical mappings. It is shown that the resulting signal can be a random process described in terms of fractal distributions and fractal domain integrals. In typical cases a steady state satisfies a dilatation equation, relating an unknown function $f(x)$ to $f(\kappa x)$ (for example, ${f(x)=g(x)f(\kappa x)}$). We discuss simple linear models as well as nonlinear systems with chaotic behavior including dissipative circuits with delayed feedback.